User blog:Denis Maksudov/Googology of Archimedes
Earliest googological notation in human history was invented by Archimedes (c. 287 – c. 212 BC) with aim to calculate how much grains of sand the Universe can contain (in supposition, that all universe filled by sand). Method of calculation of grains Greek bold text, which you can see below, was copied from The Sand Reckoner, translation in my words. Archimedes used the following assumptions: 1) περίμετρον τᾶς γᾶς εἶμεν ὡς τʹ μυριάδων σταδίων - The perimeter of the Earth is not bigger than 3 000 000 stadia (555 000 km) - Archimedes specially greatly overstates estimation of earth circumference. 2) διάμετρος τοῦ ἁλίου ἐλάττων ἐστὶν ἢ τριακονταπλασίων τᾶς διαμέτρου τᾶς γᾶς - Diameter of the Sun no more than 30 times bigger than diameter of the Earth (Archimedes used heliocentric model of Aristarchus). As Archimedes showed with help of his measurement of angular diameter of the Sun, Diameter of the Earth orbit no more than 300 times bigger than diametr of the Sun. Consequently, diameter of the Earth orbit no more than 10 000 times bigger than diameter of the Earth 3)διάμετρος τᾶς τῶν ἁπλανέων ἄστρων σφαίρας ἐλάσσων ἐστὶν ἢ μυριοπλασίων τᾶς διαμετρου του κοσμου - Diameter of sphere of unmoving stars (Diameter of universe) no more than 10 000 times bigger than diameter of the Earth orbit Thus \(D_{universe}/D_{Earth \quad orbit}=D_{Earth \quad orbit}/D_{Earth} \leq 10^4\) and \(D_{Earth} \leq 10^6\) stadia. It is easy to calculate, that in this case size of universe \(\leq 10^{14} \) stadia (about 2 light years) 4) Diameter of sand grains no less than 1/10 000 000 stadia Thus, the full volume of the universe no more than N times bigger than volume of a sand grain, where \(N=(10^7\times10^6 \times 10^4 \times 10^4)^3=10^{63}\) It was very interesting for me, how could Archimedes write numbers up to \(10^{63}\) using only Greek numerals, which you can see in table below Greek Numerals Example: ͵θτπεʹ=9385 1-μονὰς, 10-δέκα, 100-ἑκατόν, 1000-χιλιάς, 10000-μύριον (myriad) Ten thousand (one myriad) ancient greeks wrote as "M" and quantity of myriads they wrote before "M", for example ͵θτπεM͵θτπεʹ=93 859 385 Thus, maximal number, which ancient greeks could write was ,θϡϟθM,θϡϟθ' = 99 999 999 and this was not enough for mentioned calculation. That is why Archimedes had to invent his own notation. Notation of Archimedes ἀριθμοὶ ἐς τὰς μυρίας μυριάδας πρώτοι καλουμένοι natural numbers up to myriad of myriads \((10^{8})\) are numbers of first order πρώτων ἀριθμῶν αἱ μυρίαι μυριάδες μονὰς καλείσθω δευτέρων ἀριθμῶν the myriad of myriads of first order is equal to the unit of second order \((10^{8})\) μυρίαι μυριάδες τῶν δευτέρων ἀριθμῶν μονὰς καλείστω τρίτων ἀριθμῶν the myriad of myriads of second order is equal to the unit of third order \((10^{8}\times 10^{8}=10^{16})\) τρίτων ἀριθμῶν μυρίαι μυριάδες μονὰς καλείσθω τετάρτων ἀριθμῶν the myriad of myriads of third order is equal to the unit of fourth order \((10^{16}\times 10^{8}=10^{24})\) τετάρτων ἀριθμῶν μυρίαι μυριάδες μονὰς καλείσθω πέμπτων ἀριθμῶν the myriad of myriads of fourth order is equal to the unit of fifth order \((10^{24}\times 10^{8}=10^{32})\) and so on up to μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας myriad of myriads of 100 000 000-th order \((10^{8\times 10^8})\) Examples ιʹ μονάδες τῶν δευτέρων ἀριθμῶν = ten of units of second order \(=10\times 10^{8\times(2-1)}= 10^{9}\) μυρίαι μυριάδες τῶν δευτέρων ἀριθμῶν = myriad of myriads of second order=αʹ μονάδα τῶν τρίτων ἀριθμῶν = one unit of third order \(=10^{8\times(3-1)}= 10^{16}\) ιʹ μυριάδες τῶν τρίτων ἀριθμῶν = ten myriads of third order \(=10\times10^4\times10^{8\times(3-1)}= 10^{21}\) ιʹ μονάδες τῶν πέμπτων ἀριθμῶν = ten of units of fifth order \(=10\times 10^{8\times(5-1)}= 10^{33}\) ιʹ μυριάδες τῶν ἕκτων ἀριθμῶν = ten myriads of sixth order \(=10\times10^4\times10^{8\times(6-1)}=10^{45}\) ͵α μονάδες τῶν ἑβδόμων ἀριθμῶν = thousand of units of seventh order \(=10^3\times10^{8\times(7-1)}=10^{51}\) = χιλίαι μονάδες τῶν ἑβδόμων ἀριθμῶν ͵α μυριάδες τῶν ὀγδόων ἀριθμῶν = thousand of myriads of eighth order = \(=10^3\times10^4\times10^{8\times(8-1)}=10^{63}\) ͵θτπεʹ μονάδες τῶν μυριακισμυριοστῶν ἀριθμῶν = 9385 units of 100 000 000-th order = \(=9385 \times 10^{799 999 992}\) In general case each number up to \(10^{800 000 000}\) can be uniquely written in next form \(N=\sum_{i=1}^k\) (Number of myriads of i-th order+Number of units of i-th order) Extension Let all mentioned numbers are the numbers of first period (ἀριθμοὶ πρώτας περιόδου ) ὁ δὲ ἔσχατος ἀριθμὸς τᾶς πρώτας περιόδου μονὰς καλείσθω δευτέρας περιόδου πρώτων ἀριθμῶν And last number of first period \((10^{8\times 10^8})\) is unit of first order of second period μυρίαι μυριάδες τᾶς δευτέρας περιόδου πρώτων ἀριθμῶν μονὰς καλείσθω τᾶς δευτέρας περιόδου δευτέρων ἀριθμῶν the myriad of myriads of first order of second period is equal to the unit of second order of second period \(=10^{8\times 10^8 +8}\) \(10^{16\times 10^8}\) is unit of first order of third period \(10^{24\times 10^8}\) is unit of first order of fourth period \(10^{32\times 10^8}\) is unit of first order of fifth period and so on up to μυριακισμυριοστᾶς περιόδου μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας = = myriad of myriads of 100 000 000-th order of 100 000 000-th period \(=10^{8\times 10^{16}}\) And this is the limit of the notation Category:Blog posts